1
The Benefits of Coding over Routing in a
Randomized Setting
Tracey Ho, Ralf Koetter, Muriel Médard, David R. Karger and Michelle Effros
Abstract— We present a novel randomized coding approach for robust, distributed transmission and compression of information in networks. We give a lower bound on
the success probability of a random network code, based on
the form of transfer matrix determinant polynomials, that
is tighter than the Schwartz-Zippel bound for general polynomials of the same total degree. The corresponding upper
bound on failure probability is on the order of the inverse
of the size of the finite field, showing that it can be made
arbitrarily small by coding in a sufficiently large finite field,
and that it decreases exponentially with the number of codeword bits.
We demonstrate the advantage of randomized coding
over routing for distributed transmission in rectangular
grid networks by giving, in terms of the relative grid locations of a source-receiver pair, an upper bound on routing success probability that is exceeded by a corresponding
lower bound on coding success probability for sufficiently
large finite fields.
We also show that our lower bound on the success probability of randomized coding holds for linearly correlated
sources. This implies that randomized coding effectively
compresses linearly correlated information to the capacity
of any network cut in a feasible connection problem.
I. I NTRODUCTION
In this paper we present a novel randomized coding approach for robust, distributed transmission and compression of information in networks, and demonstrate its advantages over routing-based approaches.
It is known that there exist cases where coding over
networks enables certain connections that are not possible with just routing [1]. In this paper we investigate the
benefits of coding over routing, not in terms of a taxonomy of network connection problems for which coding is
necessary, but in a probabilistic, distributed setting. Distributed randomized routing has previously been considTracey Ho and Muriel Médard are with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology,
Cambridge, MA 02139, e-mail: {trace, medard}@mit.edu
Ralf Koetter is with the Coordinated Science Laboratory, University
of Illinois, Urbana, IL 61801, e-mail: [email protected]
David R. Karger is with the Laboratory for Computer Science, Massachusetts Institute of Technology, MA 02139, e-mail:
[email protected]
Michelle Effros is with the Data Compression Laboratory, California Institute of Technology, Pasadena, CA 91125, e-mail:
[email protected]
ered for achieving robustness and path diversity with minimal state [5].
We give a lower bound on the success probability of
a random network code, based on the form of transfer
matrix determinant polynomials, that is tighter than the
Schwartz-Zippel bound for general polynomials of the
same total degree. The corresponding upper bound on
failure probability is on the order of the inverse of the size
of the finite field, showing that it can be made arbitrarily
small by coding in a sufficiently large finite field, and that
it decreases exponentially with the number of codeword
bits. This suggests that random codes are potentially very
useful for networks with unknown or changing topologies.
We demonstrate the advantage of randomized coding over routing for distributed transmission of multiple
source processes in the case of rectangular grid networks.
We provide an upper bound on the routing success probability for a source-receiver pair in terms of their relative
grid locations, which is surpassed by the corresponding
lower bound for randomized coding in sufficiently large
finite fields.
Randomized coding also has connections with distributed data compression. We show that our lower bound
on the success probability of randomized coding applies
also for linearly correlated sources, which arise naturally
in applications such as networks of sensors measuring the
additive effects of multiple phenomena and noise. The effect of randomized coding on such sources can be viewed
as distributed compression occuring within the network
rather than at the sources. For a feasible multicast connection problem (i.e., one for which there exists some coding
solution) and a randomized code of sufficient complexity, with high probability the information flowing across
any cut will be sufficient to reconstruct the original source
processes. In effect, the source information is being compressed to the capacity of any cut that it passes through.
This is achieved without the need for any coordination
among the source nodes, which is advantageous in distributed environments where such coordination is impossible or expensive.
Finally, we note that this randomized coding approach
achieves robustness in a way quite different from traditional approaches. Traditionally, compression is applied
at source nodes so as to minimize required transmission
2
rate and leave spare network capacity, and the addition of
new sources may require re-routing of existing connections. Our approach fully utilizes available or allocated
network capacity for maximal robustness, while retaining
full flexibility to accommodate changes in network topology or addition of new sources.
The paper is organized as follows: Section II describes
our network model, Section III gives the main results,
Section IV gives proofs and ancillary results, and Section V concludes the paper with a summary of the results
and a discussion of further work.
II. M ODEL
We adopt the model of [3], which represents a network
as a directed graph G. Discrete independent random processes X1 , . . . , Xr are observable at one or more source
nodes, and there are d ≥ 1 receiver nodes. The output
processes at a receiver node β are denoted Z(β, i). The
multicast connection problem is to transmit all the source
processes to each of the receiver nodes.
There are ν links in the network. Link l is an incident
outgoing link of node v if v = tail(l), and an incident incoming link of v if v = head(l). We call an incident outgoing link of a source node a source link and an incident
incoming link of a receiver node a terminal link. Edge l
carries the random process Y (l).
The time unit is chosen such that the capacity of each
link is one bit per unit time, and the random processes Xi
have a constant entropy rate of one bit per unit time. Edges
with larger capacities are modelled as parallel edges, and
sources of larger entropy rate are modelled as multiple
sources at the same node.
The processes Xi , Y (l), Z(β, i) generate binary sequences. We assume that information is transmitted as
vectors of bits which are of equal length u, represented as
elements in the finite field F2u . The length of the vectors
is equal in all transmissions and all links are assumed to
be synchronized with respect to the symbol timing.
In this paper we consider linear coding1 on acyclic
delay-free networks2 . In a linear code, the signal Y (j)
on a link j is a linear combination of processes Xi generated at node v = tail(j) and signals Y (l) on incident
incoming links l:
and an output process Z(β, i) at receiver node β is a linear
combination of signals on its terminal links:
bβ i,l Y (l)
Z(β, i) =
{l : head(l)=β}
The coefficients {ai,j , fl,j , bβ i,l ∈ F2u } can be collected into r × ν matrices A = (ai,j ) and Bβ = (bβ i,j ),
and the ν × ν matrix F = (fl,j ), whose structure is constrained by the network. A triple (A, F, B), where


B1

:
B=
Bd
specifies the behavior of the network, and represents a linear network code. We use the following notation:
• G = (I − F )−1
• GH is the submatrix consisting of columns of G corresponding to links in set H
• aj , cj and bj denote column j of A, AG and B respectively
III. M AIN RESULTS
Reference [3] gives an algorithm for finding a linear coding solution to a given multicast problem, using
knowledge of the entire network topology. In applications where communication is limited or expensive, it
may be necessary or useful to determine each node’s behavior in a distributed manner. We consider a randomized approach in which network nodes independently and
randomly choose code coefficients from some finite field
Fq . The only management information needed by the receivers is the overall linear combination of source processes present in each of their incoming signals. This information can be maintained, for each signal in the network, as a vector in Fq r of the coefficients of each of the
source processes, and updated by each coding node applying the same linear combinations to the coefficient vectors
as to the data.
Our first result gives a lower bound on the success rate
of randomized coding over Fq , in terms of the number
of receivers and the number of links in the network. Because of the particular form of the product of transfer matrix determinant polynomials, the bound is tighter than the
Schwartz-Zippel bound of of dν/q for general polynomiai,j Xi +
fl,j Y (l) als of the same total degree.
Y (j) =
{l : head(l) = v}
{i : Xi generated at v}
Theorem 1: For a feasible multicast connection problem
with independent or linearly correlated sources, and
1
which is sufficient for multicast [4]
a
network
code in which some or all code coefficients are
2
this algebraic framework can be extended to networks with cycles
and delay by working in fields of rational functions in a delay vari- chosen independently and uniformly over all elements of
a finite field Fq (some coefficients can take fixed values as
able [3]
3
long as these values preserve feasibility3 ), the probability
that all the receivers can decode the source processes is at
least (1 − d/q)ν for q > d, where d is the number of receivers and ν is the maximum number of links receiving
signals with independent randomized coefficients in any
set of links constituting a flow solution from all sources to
any receiver.
The complexity of the code grows as the logarithm of
the field size q = 2u , since arithmetic operations are performed on codewords of length u. The bound is on the
order of the inverse of the field size, so the error probability decreases exponentially with the number of codeword
bits u. For a fixed success probability, the field size needs
to be on the order of the number of links ν multiplied by
the number of receivers d.
An implication of this result for linearly correlated
sources is that for a feasible multicast connection problem
and a randomized code of sufficient complexity, with high
probability the information passing through any sourcereceiver cut in the network contains the source information in a form that is compressed (or expanded) to the capacity of the cut.
Unlike random coding, if we consider only routing solutions (where different signals are not combined), then
there are network connection problems for which the success probability of distributed routing is bounded away
from 1.
Consider for example the problem of sending two processes from a source node to receiver nodes in random
unknown locations on a rectangular grid network. Transmission to a particular receiver is successful if the receiver
gets two different processes instead of duplicates of the
same process. Suppose we wish to use a distributed transmission scheme that does not involve any communication
between nodes or routing state (perhaps because of storage or complexity limitations of network nodes, or frequent shifting of receiver nodes). The best the network
can aim for is to maximize the probability that any node
will receive two distinct messages, by flooding in a way
that preserves message diversity, for instance using the
following scheme RR (ref Figure 1):
• The source node sends one process in both directions
on one axis and the other process in both directions
along the other axis.
• A node receiving information on one link sends the
same information on its three other links (these are
nodes along the grid axes passing through the source
node).
• A node receiving signals on two links sends one of
3
i.e. the result holds for networks where not all nodes perform random coding, or where signals add by superposition on some channels
the incoming signals on one of its two other links
with equal probability, and the other signal on the
remaining link.
11
00
00
11
00
11
1
0
0
1
0
1
1
0
0
1
0
1
11
00
00
11
00
11
1
0
0
1
0
1
11
00
00
11
1
0
0
1
00
11
11
00
00
11
0
1
1
0
0
1
X2
11
00
00
11
1
0
0
1
X1
1
0
0Src
1
X1
X2
00
11
11
00
00
11
0
1
1
0
0
1
0
1
1
0
0
1
Fig. 1. Rectangular grid network.
Theorem 2: For the random routing scheme RR, the
probability that a receiver located at grid position (x, y)
relative to the source receives both source processes is at
most
1 + 2||x|−|y||+1 (4min(|x|,|y|)−1 − 1)/3
2|x|+|y|−2
For comparison, we consider the same rectangular grid
problem with the following simple random coding scheme
RC (ref Figure 1):
• The source node sends one process in both directions
on one axis and the other process in both directions
along the other axis.
• A node receiving information on one link sends the
same information on its three other links.
• A node receiving signals on two links sends a random
linear combination of the source signals on each of
its two other links.4
Theorem 3: For the random coding scheme RC, the
probability that a receiver located at grid position (x, y)
relative to the source can decode both source processes is
at least (1 − 1/q)2(x+y−2) .
Table III gives, for various values of x and y, the values
of the success probability bounds as well as some actual
probabilities for routing when x and y are small. Note that
an increase in grid size from 3 × 3 to 10 × 10 requires only
an increase of two in codeword length to obtain success
probability lower bounds close to 0.9, which are substantially better than the upper bounds for routing.
IV. P ROOFS AND ANCILLARY RESULTS
We make use of the following result from our companion paper [2], which characterizes the feasibility of a multicast connection problem in terms of network flows:
4
This simple scheme, unlike the randomized routing scheme RR,
leaves out the optimization that each node receiving two linearly independent signals should always send out two linearly independent
signals.
4
TABLE I
S UCCESS PROBABILITIES OF RANDOMIZED ROUTING SCHEME RR AND RANDOMIZED CODING SCHEME RC
Receiver position
actual
RR
upper bound
F24 lower bound
RC F26 lower bound
F28 lower bound
(2,2)
0.75
0.75
0.772
0.939
0.984
(3,3)
0.672
0.688
0.597
0.881
0.969
(4,4)
0.637
0.672
0.461
0.827
0.954
(10,10)
0.667
0.098
0.567
0.868
Theorem 4: A multicast connection problem is feasible
(or a particular (A, F ) can be part of a valid solution) if
and only if each receiver β has a set Hβ of r incident
incoming links for which
PHβ =
A{l
1 ,...,lr
{disjoint paths E1 , . . . , Er :
Ei from outgoing link
li of source i to hi ∈ Hβ }
r
g(Ej ) = 0
}
j=1
where A{l1 ,...,lr } is the submatrix of A consisting of
columns corresponding to links {l1 , . . . , lr }. The sum is
over all flows that transmit all source processes to links in
Hβ , each flow being a set of r disjoint paths each connecting a different source to a different link in Hβ .
Corollary 1: The polynomial Pβ for each receiver has
maximum degree ν and is linear in variables {ax,j , fi,j }.
The product of d such polynomials has maximum degree
dν, and the largest exponent of any variable {ax,j , fi,j } is
at most d.
The particular form given in Corollary 1 of the product
of determinant polynomials gives rise to a tighter bound
on its probability of being zero when its variables take
random values from a finite field Fq , as compared to the
Schwartz-Zippel bound of dν/q for a general dν-degree
multivariate polynomial.
Lemma 1: Let P be a polynomial of degree less than or
equal to dν, in which the largest exponent of any variable
is at most d. The probability that P equals zero is at most
1 − (1 − d/q)ν for d < q.
Proof: For any variable ξ1 in P , let d1 be the largest
exponent of ξ1 in P . Express P in the form P = ξ1d1 P1 +
R1 , where P1 is a polynomial of degree at most dν − d1
that does not contain variable ξ1 , and R1 is a polynomial
in which the largest exponent of ξ1 is less than d1 . By the
Principle of Deferred Decisions, the probability Pr[P =
0] is unaffected if we set the value of ξ1 last after all the
other coefficients have been set. If, for some choice of the
other coefficients, P1 = 0, then P becomes a polynomial
of degree d1 in ξ1 . By the Schwartz-Zippel Theorem, this
probability Pr[P = 0|P1 = 0] is upper bounded by d1 /q.
So
Pr[P = 0] ≤ Pr[P1 = 0]
d1
+ Pr[P1 = 0]
q
(2,3)
0.562
0.625
0.679
0.910
0.977
(9,10)
0.667
0.111
0.585
0.875
(2,4)
0.359
0.563
0.597
0.882
0.969
(8,10)
0.667
0.126
0.604
0.882
d1
d1
= Pr[P1 = 0] 1 −
+
q
q
(1)
Next we consider Pr[P1 = 0], choosing any variable ξ2
in P1 and letting d2 be the largest exponent of ξ2 in P1 .
We express P1 in the form P1 = ξ2d2 P2 + R2 , where P2
is a polynomial of degree at most dν − d1 − d2 that does
not contain variable ξ2 , and R2 is a polynomial in which
the largest exponent of ξ2 is less than d2 . Proceeding
similarly, we assign variables ξi and define di and Pi for
i = 3, 4, . . . until we reach i = k where Pk is a constant
and
k Pr[Pk = 0] = 0. Note that 1 ≤ di ≤ d < q ∀ i and
i=1 di ≤ dν, so k ≤ dν. Applying Schwartz-Zippel as
before, we have for k = 1, 2, . . . , k
dk +1
dk +1
+
P r[Pk = 0] ≤ P r[Pk +1 = 0] 1 −
q
q
(2)
Combining all the inequalities recursively, we can show
by induction that
k
d
i
i=j di dj
i=1
Pr[P = 0] ≤
−
+ ...
q
q2
k
k−1
i=1 di
+(−1)
qk
where 0 ≤ dν − ki=1 di .
Now consider the integer optimization problem
dν
i=j di dj
i=1 di
−
+ ...
Maximize f =
q
q2
dν
dν−1
i=1 di
+(−1)
q dν
subject to 0 ≤ di ≤ d < q ∀ i ∈ [1, dν],
dν
di ≤ dν, and di integer
(3)
i=1
whose maximum is an upper bound on Pr[P = 0].
We first consider the non-integer relaxation of the problem. Let d∗ = {d∗1 , . . . , d∗dν } be an optimal solution.
of h distinct
For any set Sh integers from [1, dν],
let fSh = 1 −
i∈Sh
q
di
+
i,j∈Sh ,i=j
q2
di dj
− ... +
5
i∈S
di
(−1)h qhh . We can show by induction on h that
0 < fSh < 1 for any set Sh of h distinct integers in [1, dν].
∗
∗
If dν
i=1 di < dν, then there is some di < d, and there
exists a feasible solution d such that di = d∗i + %, % > 0,
and dh = d∗h for h = i, which satisfies
∗
%
h=i dh
∗
1−
+
f (d) − f (d ) =
q
q
∗
h=i dh
. . . + (−1)dν−1 dν−1
q
This is positive, contradicting the optimality of d∗ .
Next suppose 0 < d∗i < d for some d∗i . Then there
exists some d∗j such that 0 < d∗j < d, since if d∗j = 0 or d
∗
for all other j, then dν
i=1 di = dν. Assume without loss
∗
of generality that 0 < di ≤ d∗j < d. Then there exists a
feasible vector d such that di = d∗i −%, dj = d∗j +%, % > 0,
and dh = d∗h ∀ h = i, j, which satisfies
∗ − d∗ )% − %2
(d
i
j
f (d) − f (d∗ ) = −
q2
∗
∗
h=i,j dh
h=i,j dh
dν−2
− . . . + (−1)
1−
q
q dν−2
∗
This is again
contradicting the optimality of d .
dνpositive,
∗ = dν, and d∗ = 0 or d. So exd
Thus,
i
i=1 i
actly ν of the variables d∗i are equal to d. Since the optimal solution is an integer solution, it is also optimal
for the integer program (3). The corresponding
ν
optimal
ν d2
dν
d
d
ν−1
=
1
−
1
−
.
f = ν q − 2 q2 + . . . + (−1)
ν
q
q
Proof of Theorem 1: By Corollary 1, the product
P
β β has degree at most dν, and the largest exponent
of any variable ax,j or fi,j is at most d. These properties
still hold if some variables are set to deterministic values
which do not make the product identically zero.
Linearly correlated sources can be viewed as prespecified linear combinations of underlying independent
processes. Unlike the independent sources case where
each nonzero entry of the A matrix can be set independently, in this case there are linear dependencies among
the entries. The columns aj of the A matrix are linear
functions aj = k αjk v kj of column vectors v kj that represent the composition of the source processes at tail(j)
in terms of the underlying independent processes: Variables αjk in column aj can be set independently of variables αjk in other columns aj . It can be seen from Theorem 4 that for any particular j, each product term in the
polynomial Pβ for
any kreceiver β contains at most one
. Pβ is thus linear in the varivariable ai,j = k αjk vi,j
k
ables αj , and also in variables fi,j , which are unaffected
by the source correlations. So any variable in the product
of d such polynomials has maximum exponent d.
Applying Lemma 1 gives us the required bound.
For the single-receiver case, the bound is attained for a
network consisting only of links forming a single set of r
disjoint source-receiver paths.
Proof of Theorem 2: To simplify notation, we assume without loss of generality that the axes are chosen
such that the source is at (0, 0), and 0 < x ≤ y. Let Ex,y
be the event that two different signals are received by a
node at grid position (x, y) relative to the source. The
statement of the lemma is then
Pr[Ex,y ] ≤ 1 + 2y−x+1 (4x−1 − 1)/3 /2y+x−2
(4)
which we prove by induction.
h denote the signal carried on the link between
Let Yx,y
v denote the signal carried
(x − 1, y) and (x, y) and let Yx,y
on the link between (x, y − 1) and (x, y) (ref Figure 2).
h
00
y11
00
11
11
00
00
11
Y x−1,y
11
00
00
11
11
00
00
11
v
Y x−1,y
h
Y x, y−1
00
11
y−1
00
11
11
00
00
11
11
00
00
11
11
00
00
11
00
11
00
11
11
00
00
11
00
11
11
00
00
11
00
11
11
00
00
11
11
00
00
11
11
00
00
11
11
00
00
11
v
Y x,y−1
00
11
y−2
00
11
11
00
00
11
111111111111111111111
000000000000000000000
x−2
x−1
x
h
Fig. 2. Rectangular grid network. Yx,y
denotes the signal carried
v
denotes the signal
on the link between (x − 1, y) and (x, y), and Yx,y
carried on the link between (x, y − 1) and (x, y).
Observe that Pr[Ex,y |Ex−1,y ] = 1/2, since with probability 1/2 node (x − 1, y) transmits to node (x, y) the signal complementary to whatever signal is being transmitted
from node (x, y − 1). Similarly, Pr[Ex,y |Ex,y−1 ] = 1/2,
so Pr[Ex,y |Ex−1,y or Ex,y−1 ] = 1/2.
Case 1: Ex−1,y−1
h
v
v
h
Case 1a: Yx−1,y
= Yx,y−1
. If Yx−1,y
= Yx−1,y
,
v
h
then Ex,y−1 ∪ Ex−1,y , and if Yx,y−1 = Yx,y−1 ,
then E x,y−1 ∪ E x−1,y . So Pr[Ex,y | Case 1a] =
1
1
1
3
2 × 2 + 2 = 4.
h
v
= Yx,y−1
. Either Ex,y−1 ∪ E x−1,y
Case 1b: Yx−1,y
or E x,y−1 ∪ Ex−1,y , so Pr[Ex,y | Case 1b] = 1/2.
Case 2: E x−1,y−1
h
v
Case 2a: Yx−1,y
= Yx,y−1
. Either Ex,y−1 ∪ E x−1,y
or E x,y−1 ∪ Ex−1,y , so Pr[Ex,y | Case 2a] = 1/2.
h
v
h
Case 2b: Yx−1,y
= Yx,y−1
= Yx−1,y−1
. By the asv
sumption of case 2, Yx,y−1 is also equal to this same
signal, and Pr[Ex,y | Case 2b] = 0.
6
h
v
h
Case 2c: Yx−1,y
= Yx,y−1
= Yx−1,y−1
. Then Ex,y−1
and Ex−1,y , so Pr[Ex,y | Case 2c] = 1/2.
So
Pr[Ex,y |Ex−1,y−1 ] ≤ max (Pr[Ex,y | Case 1a],
Pr[Ex,y | Case 1b]) = 3/4
Pr[Ex,y |E x−1,y−1 ] ≤ max (Pr[Ex,y | Case 2a],
Pr[Ex,y | Case 2b], Pr[Ex,y | Case 2c]) = 1/2
3
1
Pr[Ex,y ] ≤ Pr[Ex−1,y−1 ] + Pr[E x−1,y−1 ]
4
2
1 1
= + Pr[Ex−1,y−1 ]
2 4
If Equation 4 holds for some (x, y), then it also holds
for (x + 1, y + 1):
1 1
Pr[Ex+1,y+1 ] ≤ + Pr[Ex,y ]
2 4
1 1 1 + 2y−x+1 (1 + 4 + . . . + 4x−2 )
= +
2 4
2y+x−2
y−x+1
x
(4 − 1)/3
1+2
=
y+1+x+1−2
2
Now Pr[E1,y ] = 1/2y −1 , since there are y − 1 nodes,
(1, 1), . . . , (1, y − 1), at which one of the signals is eliminated with probability 1/2. Setting y = y − x + 1 gives
the base case which completes the induction.
Proof of Theorem 3: We first establish the degree
of the transfer matrix determinant polynomial Pβ for a receiver β at (x, y), in the indeterminate variables fi,j . By
Theorem 4, Pβ is a linear combination of product terms
of the form a1,l1 a2,l2 fi1 ,l3 . . . fil ,lk , where {l1 , . . . , lk } is
a set of distinct links forming two disjoint paths from the
source to the receiver. In the random coding scheme we
consider, the only randomized variables are the fi,j variables at nodes receiving information on two links. The
maximum number of such nodes on a source-receiver path
is x + y − 2, so the total degree of Pβ is 2(x + y − 2). Applying the random coding bound of Lemma 1 yields the
result.
V. C ONCLUSIONS AND FURTHER WORK
We have presented a novel randomized coding approach for robust, distributed transmission and compression of information in networks, giving an upper bound
on failure probability that decreases exponentially with
codeword length. We have demonstrated the advantages
of randomized coding over randomized routing in rectangular grid networks, by giving an upper bound on the
success probability of a randomized routing scheme that
is exceeded by the corresponding lower bound for a simple randomized coding scheme in sufficiently large finite
fields.
We have also shown that randomized coding has the
same success bound for linearly correlated sources, with
the implication that randomized coding effectively compresses correlated information to the capacity of any cut
that it passes through.
Finally, we note that this randomized coding approach offers a new paradigm for achieving robustness,
by spreading information over available network capacity while retaining maximum flexibility to accommodate
changes in the network.
Several areas of further research spring from this work.
One such area is to study more sophisticated randomized coding schemes on various network topologies, and
to compare their performance and management overhead
with that of deterministic schemes. Another area would be
to extend our results to sources with arbitrary correlations
and networks with cycles and delay.
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